Random affine simplexes

For a fixed $k\in \{1,\dots ,d\}$ consider random vectors $X_0,\dots, X_{k}\in\mathbb R^d$ with an arbitrary spherically symmetric joint density function. Let $A$ be any non-singular $d\times d$ matrix. We show that the $k$-dimensional volume of the convex hull of affinely transformed $X_i$'s satisfies $$\mathrm{\mid }\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(A{X}_0,\dots ,A{X}_k)\mathrm{\mid }\frac{\mathrm{\mid }{P}_{\xi }\mathcal{E}\mathrm{\mid }}{{\kappa }_k}\cdot \mathrm{\mid }\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(X_0,\dots ,X_k)\mathrm{\mid },$$ where $$\mathcal{E}$:={$\mathbf{x}$\in\mathbb R^d:{$\mathbf{x}$^\top (A^\top A)^{-1}$\mathbf{x}$}\leq 1}$isanellipsoid,$P_\xi$ denotes the orthogonal projection to a random uniformly chosen $k$-dimensional linear subspace $\xi$ independent of $X_0,\dots ,X_k$, and ${\kappa }_k$ is the volume of the unit $k$-dimensional ball. We express $\mathrm{\mid }{P}_{\xi }\mathcal{E}\mathrm{\mid }$ in terms of Gaussian random matrices. The important special case $k=1$ corresponds to the distance between two random points: $$\mathrm{\mid }A{X}_0-A{X}_1\mathrm{\mid }\sqrt{\frac{{\lambda }_1^2{N}_1^2+\cdots +{\lambda }_d^2{N}_d^2}{N_1^2+\cdots +N_d^2}}\cdot \mathrm{\mid }{X}_0-X_1\mathrm{\mid },$$ where $N_1,\dots ,N_d$ are i.i.d. standard Gaussian variables independent of $X_0,X_1$ and ${\lambda }_1,\dots ,{\lambda }_d$ are the singular values of $A$. As an application, we derive the following integral geometry formula for ellipsoids: $$\frac{{\kappa }_d^{k+1}}{{\kappa }_k^{d+1}}\frac{{\kappa }_{k(d+p)+k}}{{\kappa }_{k(d+p)+d}}\underset{A_{d,k}}{\int }\mathrm{\mid }\mathcal{E}\cap E{\mathrm{\mid }}^{p+d+1}{\mu }_{d,k}(dE)=\mathrm{\mid }\mathcal{E}{\mathrm{\mid }}^{k+1}\underset{G_{d,k}}{\int }\mathrm{\mid }{P}_L\mathcal{E}{\mathrm{\mid }}^p{\nu }_{d,k}(dL),$$ where $p>-d+k-1$ and $A_{d,k}$ and $G_{d,k}$ are the affine and the linear Grassmannians equipped with their respective Haar measures. The case $p=0$ reduces to an affine version of the integral formula of Furstenberg and Tzkoni.